The integral you are working with is:

$\int_{-4}^{8} (4 - 5x) \, dx$

This is a definite integral, and we can evaluate it step by step.

Step 1: Find the antiderivative

The function to integrate is $4 - 5x$. The antiderivative (or indefinite integral) of each term is calculated as follows:

• The antiderivative of $4$ is $4x$.
• The antiderivative of $-5x$ is $-\frac{5}{2}x^2$.

Thus, the antiderivative of $4 - 5x$ is:

$4x - \frac{5}{2}x^2$

Step 2: Apply the limits of integration

Now, we need to evaluate this antiderivative from $-4$ to $8$.

First, plug in the upper limit $8$:

$\left( 4(8) - \frac{5}{2}(8)^2 \right) = 32 - \frac{5}{2}(64) = 32 - 160 = -128$

Next, plug in the lower limit $-4$:

$\left( 4(-4) - \frac{5}{2}(-4)^2 \right) = -16 - \frac{5}{2}(16) = -16 - 40 = -56$

Step 3: Subtract the values

Now subtract the value at the lower limit from the value at the upper limit:

$-128 - (-56) = -128 + 56 = -72$

Final Answer:

The exact value of the definite integral is:

$\boxed{-72}$

Would you like more details or explanations on any step?

Here are some related questions to expand on this:

1. How do you find the antiderivative of a polynomial function?
2. Why do we subtract the lower limit evaluation in definite integrals?
3. What geometric interpretation can be applied to this integral?
4. How would you solve this integral if the limits of integration were different?
5. How do you handle integrals with absolute values in the integrand?

Tip: Always double-check the sign of each term when applying the fundamental theorem of calculus, especially when subtracting values at the limits.